HopfionSeries Paper II - BPS Structure and Fixed-Point Theorem for the Density-Feedback Faddeev--Niemi Hopfion

Paper I establishes numerically that the Euler–Lagrange minimum of the density-feedback energy Efb = Kfb · J4 satisfies Kfb/J4 = λ/21/3 to within O(h2) discretisation error (+0.020% on a 256×256 lattice), where λ = φ6. This paper addresses the statement analytically in three steps, introducing the generalised functional G[f; λeff] = Kfb[f] + λeff J4[f] and the ratio map H(λeff) := Kfb[fλeff]/J4[fλeff]. Existence For each λeff > 0, G has a critical point fλeff in the Q = 2 topological sector, via a mountain-pass argument in the energy space X2 (Theorem 2.1, subject to a concentration-compactness condition resolved in Paper III via Rellich–Kondrachov compactness). Monotonicity and uniqueness H is strictly decreasing with H(0+) > 0 and H(λeff) − λeff → −∞, so the fixed-point equation H(λeff*) = λeff* has exactly one solution (Theorem 3.1, subject to a non-degeneracy condition). Fixed-point value In the thin-torus approximation, restricting to the Battye–Sutcliffe ansatz f = 2 arctan(C/d), Beta-function integration gives I4/I2a = 3/(4C2) exactly, from which self-consistency gives I4/I2a|C=C* = 24/3/φ5 (Theorem 4.2, unconditional). For the full 2D EL equation the same fixed-point value is the Linking Scale Conjecture, confirmed numerically to 0.020% and proved completely in Paper III. The thin-torus proof of this paper is the result used by Paper I to promote its conditional predictions to unconditional status. The Linking Scale Conjecture is stated precisely and its algebraic context — including S-matrix self-duality S1/2,1/2 = S0,0, twist pairing θ0 θ1/2 = 1, and fusion count Nfus = 2 for all k ≥ 2 — is established as the foundation for the full 2D proof in Paper III.